arXiv subjects

Tristan Pryer

Publications and source records attributed to Tristan Pryer.

At least 19 recordsLinked to original sources

arXiv2026

Batched Wavefront Sweeps for Linear Transport Uncertainty Quantification

We develop a graph-compatible batched wavefront formulation for upwind discontinuous Galerkin discretisations of linear transport. For sweepable discretisations, the cell equations form a block lower-triangular system under a topological ordering of the directed upwind dependency graph. This causal structure can be shared across families of fixed-source problems whose local operators, sources and inflow data vary. We exploit this observation by grouping channels with a common dependency graph into graph-compatible sweep classes and carrying samples, right-hand sides, energy groups and sign-compatible angular ordinates as tensor dimensions within a common wavefront schedule. We prove that the resulting batched wavefront algorithm is algebraically equivalent to independent classical DG sweeps for every channel in a class, while exposing parallelism simultaneously across wavefront cells and channel dimensions. We realise the formulation as a GPU tensor program based on batched cell-local solves and quantify its work, depth and storage requirements, including the throughput--memory tradeoff introduced by sample microbatching. We further characterise the discrete parameter-to-observable map on fixed face-sign branches and show that reverse-mode differentiation through the wavefront algorithm reproduces the corresponding discrete adjoint action. Numerical experiments verify the expected DG convergence and adjoint consistency, demonstrate substantial GPU execution gains from sample batching, and illustrate the method in a material-shadowing uncertainty-quantification problem and a coupled multigroup C5G7/KAIST-inspired power-iteration workload.

math.NA↗

arXiv2026

Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise

We analyse symmetric interior-penalty discontinuous Galerkin approximations of elliptic stochastic partial differential equations driven by spatial Gaussian white noise on quasi-uniform polytopic meshes. The low regularity of the exact random field precludes the usual SIP consistency argument. We instead restrict the same continuum white-noise functional canonically to the discontinuous polynomial space and separate the error into stochastic forcing-restriction and deterministic discretisation components. For every fixed polynomial degree $k\geq1$, we prove the sharp two-sided estimate \[ \|Y-Y_h\|_{L^2(Ξ;L^2(Ω))} \simeq h^{2-d/2}, \] giving rates $h$ in two dimensions and $h^{1/2}$ in three dimensions. A finite-dimensional lower bound establishes optimality. The analysis allows polytopic elements with arbitrarily many and arbitrarily small faces, including admissible non-convex elements. In two dimensions we also treat polygons with a single reentrant corner. Although the deterministic approximation exhibits an angle-dependent loss of regularity, the singular resolvent contribution is finite rank, and the sharp first-order stochastic rate is retained. The discontinuous structure yields an exact element-local sampler for the restricted white noise. Numerical experiments on irregular polygonal, non-convex polytopic and reentrant domains support the predicted convergence behaviour.

math.NA↗

arXiv2026

Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations

Diffusion synthetic acceleration is most effective when its diffusion correction reflects the transport discretisation that generates the iteration error. We develop this principle for high-order upwind discontinuous Galerkin discretisations of discrete-ordinates transport on polytopic meshes. From the discrete transport sweep, we derive the exact scalar correction that removes the source-iteration scalar error in one step. We prove that the associated scalar response is positive and self-adjoint, obtain an exact expression for the source-iteration convergence factor, and quantify the additional damping produced by vacuum leakage. Using the exact correction as a reference, we construct a transport-matched modified interior penalty correction whose boundary terms are inherited directly from homogeneous vacuum inflow. In the optically thick regime, the resulting MIP form approximates the exact correction with relative error proportional to the effective cell Knudsen number. This gives a strict acceleration of source iteration, with bounds uniform in mesh size, polynomial degree, and element face count for admissible polytopic meshes. Numerical experiments on Cartesian and centroidal Voronoi meshes confirm the predicted convergence and correction-operator scaling.

math.NA↗

arXiv2026

Geometry, Energy and Sensitivity in Stochastic Proton Dynamics

We develop numerical schemes and sensitivity methods for stochastic models of proton transport that couple energy loss, range straggling and angular diffusion. For the energy equation we introduce a logarithmic Milstein scheme that guarantees positivity and achieves strong order one convergence. For the angular dynamics we construct a Lie-group integrator. The combined method maintains the natural geometric invariants of the system. We formulate dose deposition as a regularised path-dependent functional, obtaining a pathwise sensitivity estimator that is consistent and implementable. Numerical experiments confirm that the proposed schemes achieve the expected convergence rates and provide stable estimates of dose sensitivities.

math.NA↗

arXiv2026

Diffusion Synthetic Acceleration for polytopic discretisations of Boltzmann transport

We present a computational study of diffusion synthetic acceleration (DSA) for the monoenergetic, isotropically scattering $S_N$ transport equations, discretised in space by a polytopic discontinuous Galerkin method. Using a discrete ordinates angular discretisation, we construct the DSA correction with an interior-penalty diffusion operator and compare a classical symmetric interior penalty (SIP) formulation with a modified interior penalty (MIP) variant, together with homogeneous Dirichlet and Marshak (Robin) diffusion boundary conditions imposed weakly in the DG framework. We quantify the observed convergence behaviour of the resulting source iteration across variations in optical thickness, scattering ratio, angular quadrature, mesh refinement, polynomial degree and mesh anisotropy on families of bounded Voronoi meshes. The results show that MIP-based DSA remains robust across the parameter ranges tested, whereas SIP-based DSA can lose robustness in the intermediate regime. In challenging optically thick, highly scattering settings, the observed convergence factors for the MIP-based schemes are typically below $0.6$.

math.NA↗

arXiv2026

Characteristic Sweeps and Source Iteration for Charged-Particle Transport with Continuous Slowing-Down and Angular Scattering

We develop a semi-analytic deterministic framework for charged-particle transport with continuous slowing-down in energy and angular scattering. Directed transport and energy advection are treated by method-of-characteristics integration, yielding explicit directional sweeps defined by characteristic maps and inflow data. Scattering is incorporated through a fixed-point (source-iteration) scheme in which the angular gain is lagged, yielding a sequence of decoupled directional solves coupled only through angular sums. The method is formulated variationally in a transport graph space adapted to the charged particle drift. Under standard monotonicity and positivity assumptions on the stopping power and boundedness assumptions on cross sections, we establish coercivity and boundedness of the transport bilinear form, prove contraction of the source iteration under a subcriticality condition and derive a rigorous a posteriori bound for the iteration error, providing an efficient stopping criterion. We further analyse an elastic discrete-ordinates approximation, including conservation properties and a decomposition of angular error into quadrature, cone truncation and finite iteration effects. Numerical experiments for proton transport validate the characteristic sweep against an exact ballistic benchmark and demonstrate the predicted fixed-point convergence under forward-peaked scattering. Carbon-ion simulations with tabulated stopping powers and a reduced multi-species coupling illustrate Bragg peak localisation and distal tail formation driven by secondary charged fragments.

math.NA↗

arXiv2026

Multi-level Monte Carlo Dropout for Efficient Uncertainty Quantification

We develop a multilevel Monte Carlo (MLMC) framework for uncertainty quantification with Monte Carlo dropout. Treating dropout masks as a source of epistemic randomness, we define a fidelity hierarchy by the number of stochastic forward passes used to estimate predictive moments. We construct coupled coarse--fine estimators by reusing dropout masks across fidelities, yielding telescoping MLMC estimators for both predictive means and predictive variances that remain unbiased for the corresponding dropout-induced quantities while reducing sampling variance at fixed evaluation budget. We derive explicit bias, variance and effective cost expressions, together with sample-allocation rules across levels. Numerical experiments on forward and inverse PINNs--Uzawa benchmarks confirm the predicted variance rates and demonstrate efficiency gains over single-level MC-dropout at matched cost.

cs.LG↗

arXiv2026

A finite element method preserving the eigenvalue range of symmetric tensor fields

This paper presents a finite element method that preserves (at the degrees of freedom) the eigenvalue range of the solution of tensor-valued time-dependent convection--diffusion equations. Starting from a high-order spatial baseline discretisation (in this case, the CIP stabilised finite element method), our approach formulates the fully discrete problem as a variational inequality posed on a closed convex set of tensor-valued functions that respect the same eigenvalue bounds at their degrees of freedom. The numerical realisation of the scheme relies on the definition of a projection that, at each node, performs the diagonalisation of the tensor and then truncates the eigenvalues to lie within the prescribed bounds. The temporal discretisation is carried out using the implicit Euler method, and unconditional stability and optimal-order error estimates are proven for this choice. Numerical experiments confirm the theoretical findings and illustrate the method's ability to maintain eigenvalue constraints while accurately approximating solutions in the convection-dominated regime.

math.NA↗

arXiv2025

A finite element method for a non-Newtonian dilute polymer fluid

We study the discretisation of a uniaxial (rank-one) reduction of the Oldroyd-B model for dilute polymer solutions, in which the conformation tensor is represented as $\sig = \vec b \otimes \vec b$. Building on structural analogies with MHD, we formulate a finite element framework compatible with the de Rham complex, so that the discrete velocity is exactly divergence-free. The spatial discretisation combines an interior-penalty treatment of viscosity with upwind transport to control stress layers and we prove inf-sup conditions on the mixed pairs. For time-stepping, we design an IMEX scheme that is linear at each step and show well-posedness of the fully discrete problem together with a discrete energy law mirroring the continuum dissipation. Numerical experiments on canonical benchmarks (lid-driven cavity, pipe-with-cavity and $4{:}1$ planar contraction) demonstrate accuracy and robustness for moderate-to-high Weissenberg numbers, capturing sharp stress gradients and corner singularities while retaining the efficiency gains of the uniaxial model. The results indicate that de Rham-compatible discretisations coupled with energy-stable IMEX time integration provide a reliable pathway for viscoelastic computations at elevated elasticity.

math.NA↗

arXiv2025

A posteriori analysis for nonlinear convection-diffusion systems

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

math.NA↗

arXiv2025

A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes

We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, meshes. Starting from an interior penalty discontinuous Galerkin (DG) formulation posed on a polytopic mesh, the method enforces preservation of \emph{a priori} prescribed upper and lower bounds for the numerical solution at an arbitrary number of user-defined points \emph{within} each polytopic element. This is achieved by employing a simplicial submesh and enforcing bound preservation at the submesh nodes via a nonlinear iteration. By construction, the submeshing procedure preserves the order of accuracy of the DG method, \emph{without} introducing any additional global numerical degrees of freedom compared to the baseline DG method, thereby, falling into the category of composite finite element approaches. A salient feature of the proposed method is that it automatically reverts to the standard DG method on polytopic meshes when no prescribed bound violation occurs. In particular, the choice of the discontinuity-penalisation parameter is independent of the submesh granularity. The resulting composite method combines the geometric flexibility of polytopic meshes with the accuracy and stability of discontinuous Galerkin discretisations, while rigorously guaranteeing bound preservation. The existence and uniqueness of the numerical solution is proven. A priori error bounds, assuming sufficient regularity of the exact solution are shown, employing a non-standard construction of discrete nodally bound-preserving interpolant. Numerical experiments confirm optimal convergence for smooth problems and demonstrate robustness in the presence of sharp gradients, such as boundary and interior layers.

math.NA↗

arXiv2025

Modelling and Analysis of Non-Contacting Mechanical Face Seals with Axial Disturbances and Misalignment

Advancements in industrial applications are driving developments in non-contacting mechanical seal technology. Key requirements include improvements in efficiency and reliability, which lead to smaller clearances, lower frictional losses, and minimisation of wear, during operation. However, a critical consideration is the effect of external disturbances experience by the seal from the local environment which may cause destabilisation, and lead to premature failures through unanticipated face contact. This work examines the dynamic behaviour of a non-contacting mechanical face seals, where a thin fluid film separates a pair of coaxial discs; a rotor (rotating face) and stator (stationary face). It is assumed that the rotor-stator have an angular misalignment and operation is under conditions involving large axial disturbances, representing external disturbances. A fully coupled unsteady mathematical representation is developed, where the fluid flow is coupled to the structural response of the stator, and the rotor motion is prescribed. The stator is modelled as a spring-mass-damper system, and the fluid film model is based on a lubrication approximation of the Navier-Stokes equations. The governing equations are solved via a numerical technique based on finite element and Runge-Kutta methods. A parameter study reveals the impact of misalignment on the seal dynamics, when experiencing an external disturbance. The angle of misalignment and corresponding amplitude of forcing can be identified when the minimum fluid film thickness becomes less than a given tolerance. This provides insights into safe operating conditions and manufacturing tolerances, with the research aiding to improve the design critera and reliability of non-contacting mechanical face seals.

math.NA↗

arXiv2025

Surrogate Modelling of Proton Dose with Monte Carlo Dropout Uncertainty Quantification

Accurate proton dose calculation using Monte Carlo (MC) is computationally demanding in workflows like robust optimisation, adaptive replanning, and probabilistic inference, which require repeated evaluations. To address this, we develop a neural surrogate that integrates Monte Carlo dropout to provide fast, differentiable dose predictions along with voxelwise predictive uncertainty. The method is validated through a series of experiments, starting with a one-dimensional analytic benchmark that establishes accuracy, convergence, and variance decomposition. Two-dimensional bone-water phantoms, generated using TOPAS Geant4, demonstrate the method's behavior under domain heterogeneity and beam uncertainty, while a three-dimensional water phantom confirms scalability for volumetric dose prediction. Across these settings, we separate epistemic (model) from parametric (input) contributions, showing that epistemic variance increases under distribution shift, while parametric variance dominates at material boundaries. The approach achieves significant speedups over MC while retaining uncertainty information, making it suitable for integration into robust planning, adaptive workflows, and uncertainty-aware optimisation in proton therapy.

stat.ML↗

arXiv2025

A Unified Framework from Boltzmann Transport to Proton Treatment Planning

This work develops a rigorous mathematical formulation of proton transport by integrating both deterministic and stochastic perspectives. The deterministic framework is based on the Boltzmann-Fokker-Planck equation, formulated as an operator equation in a suitable functional setting. The stochastic approach models proton evolution via a track-length parameterised diffusion process, whose infinitesimal generator provides an alternative description of transport. A key result is the duality between the stochastic and deterministic formulations, established through the adjoint relationship between the transport operator and the stochastic generator. We prove that the resolvent of the stochastic process corresponds to the Green's function of the deterministic equation, providing a natural link between fluence-based and particle-based transport descriptions. The theory is applied to dose computation, where we show that the classical relation: dose = (fluence * mass stopping power) arises consistently in both approaches. Building on this foundation, we formulate a hybrid optimisation framework for treatment planning, in which dose is computed using a stochastic model while optimisation proceeds via adjoint-based PDE methods. We prove existence and differentiability of the objective functional and derive the first-order optimality system. This framework bridges stochastic simulation with deterministic control theory and provides a foundation for future work in constrained, adaptive and uncertainty-aware optimisation in proton therapy.

math.PR↗

arXiv2025

Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries

We propose a neural network framework for solving stationary linear transport equations with inflow boundary conditions. The method represents the solution using a neural network and imposes the boundary condition via a Lagrange multiplier, based on a saddle-point formulation inspired by the classical Uzawa algorithm. The scheme is mesh-free, compatible with automatic differentiation and extends naturally to problems with scattering and heterogeneous media. We establish convergence of the continuum formulation and analyse the effects of quadrature error, neural approximation and inexact optimisation in the discrete implementation. Numerical experiments show that the method captures anisotropic transport, enforces boundary conditions and resolves scattering dynamics accurately.

math.NA↗

arXiv2025

PINN-DG: Residual neural network methods trained with Finite Elements

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics-Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation-based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second-order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness.

math.NA↗

arXiv2025

Wireless Network Topology Inference: A Markov Chains Approach

We address the problem of inferring the topology of a wireless network using limited observational data. Specifically, we assume that we can detect when a node is transmitting, but no further information regarding the transmission is available. We propose a novel network estimation procedure grounded in the following abstract problem: estimating the parameters of a finite discrete-time Markov chain by observing, at each time step, which states are visited by multiple ``anonymous'' copies of the chain. We develop a consistent estimator that approximates the transition matrix of the chain in the operator norm, with the number of required samples scaling roughly linearly with the size of the state space. Applying this estimation procedure to wireless networks, our numerical experiments demonstrate that the proposed method accurately infers network topology across a wide range of parameters, consistently outperforming transfer entropy, particularly under conditions of high network congestion.

cs.NI↗

arXiv2025

A Positivity-Preserving Finite Element Framework for Accurate Dose Computation in Proton Therapy

We present a stabilised finite element method for modelling proton transport in tissue, incorporating both inelastic energy loss and elastic angular scattering. A key innovation is a positivity-preserving formulation that guarantees non-negative fluence and dose, even on coarse meshes. This enables reliable computation of clinically relevant quantities for treatment planning. We derive a priori error estimates demonstrating optimal convergence rates and validate the method through numerical benchmarks. The proposed framework provides a robust, accurate and efficient tool for advancing proton beam therapy.

math.NA↗